Rellich type inequalities in domains of the Euclidean space

For smooth functions supported in a domain of the Euclidean space we investigate two Rellich type inequalities with weights which are powers of the distance function. We prove that for an arbitrary plane domain there exist positive Rellich constants in these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Rellich type inequalities for all non-convex domains of dimension d ≥ 2 provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres.     

Sobolev embedding theorems for a class of anisotropic irregular domains

Sufficient conditions for the embedding of a Sobolev space in Lebesgue spaces on a domain depend on the integrability and smoothness parameters of the spaces and on the geometric features of the domain. In the present paper, Sobolev embedding theorems are obtained for a class of domains with irregular boundary; this class includes the well-known classes of σ-John domains, domains with the flexible cone condition, and their anisotropic analogs.     

Log-Sobolev inequalities and regions with exterior exponential cusps

We begin by studying certain semigroup estimates which are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for proving that such log-Sobolev inequalities hold for Euclidean regions which satisfy a particular Hardy-type inequality. Our main application is to show that domains which have exterior exponential cusps, and hence have no Sobolev embedding theorem, satisfy such heat kernel bounds provided the cusps are not too sharp. Finally, we consider a rotationally invariant domain with an exponentially sharp cusp and prove that ultracontractivity breaks down when the cusp becomes too sharp.     

Polynomials,Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

This paper consists of three main parts. One of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie groups. Despite the extensive research after Jerison's work [3] on Poincaré-type inequalities for Hörmander's vector fields over the years, our results given here even in the nonweighted case appear to be new. Such interpolation inequalities have crucial applications to subelliptic or parabolic PDE's involving vector fields. The main tools to prove such inqualities are approximating the Sobolev functions by polynomials associated with the left invariant vector fields on ?. Some very usefull properties for polynomials associated with the functions are given here and they appear to have independent interests in their own rights. Finding the existence of such polynomials is the second main part of this paper. Main results of these two parts have been announced in the author's paper in Mathematical Research Letters [38].The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on (?,δ) domains. Some results of weighted Sobolev spaces are also given here. We construct a linear extension operator which is bounded on different Sobolev spaces simultaneously. In particular, we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev functions. Theorems are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.     

Spaces of functions of positive smoothness on irregular domains

The paper is devoted to constructing and studying spaces of functions of positive smoothness on irregular domains of the n-dimensional Euclidean space. We prove embedding theorems that connect the spaces introduced with the Sobolev and Lebesgue spaces. The formulations of the theorems depend on geometric parameters of the domain of definition of functions.     

Planar Poincaré domains: Geometry and Steiner symmetrization

We determine geometric necessary and sufficient conditions on a class of strip-like planar domains in order for them to satisfy the Poincaré inequality with exponentp, where 1≤p<∞. The characterization uses hyperbolic geodesics in the domain and a metric which depends onp and generalizes the quasi-hyperbolic metric in the casep=2. As an application, we show that the Poincaré inequality is preserved under Steiner symmetrization of these domains but not in general. We also show that our geometric condition is preserved under bounded length distortion (BLD) mappings of a domain and thus extend the class of domains for which our characterization is valid. The first author is supported in part by a grant from the National Science Foundation.     

Asymptotic behaviour of sharp Sobolev constants in thin domains

In this paper we study the asymptotic behaviour of the constants in Sobolev inequalities in thin domains with respect to the thickness of the domain ε. We prove that the sharp Sobolev constants in thin domains converge to the sharp Sobolev constant on the lower-dimensional domain, as ε tends to zero.     

Sobolev spaces on bounded symmetric domains

We obtain a formula for the Sobolev inner product in standard weighted Bergman spaces of holomorphic functions on a bounded symmetric domain in terms of the Peter–Weyl components in the Hua–Schmidt decomposition, and use it to clarify the relationship between the analytic continuation of these standard weighted Bergman spaces and the Sobolev spaces on bounded symmetric domains.     

拟极值距离域和拟共形映照的扩张

本文研究了空间R_n中的拟极值距离城,找到了一个可以拟共等价于球而不是拟极值距离域的例子,给出了拟极值距离域的一个等价定义,证明了一个Jordan的拟极值距离域的外部是拟极值距离城的充分条件为它是拟共形反射域。最后我们还建立了一个平面拟共形映照的扩张性质。     

Measure Density and Extension of Besov and Triebel–Lizorkin Functions

We show that a domain is an extension domain for a Haj?asz–Besov or for a Haj?asz–Triebel–Lizorkin space if and only if it satisfies a measure density condition. We use a modification of the Whitney extension where integral averages are replaced by median values, which allows us to handle also the case \(0<p<1\). The necessity of the measure density condition is derived from embedding theorems; in the case of Haj?asz–Besov spaces we apply an optimal Lorentz-type Sobolev embedding theorem which we prove using a new interpolation result. This interpolation theorem says that Haj?asz–Besov spaces are intermediate spaces between \(L^p\) and Haj?asz–Sobolev spaces. Our results are proved in the setting of a metric measure space, but most of them are new even in the Euclidean setting, for instance, we obtain a characterization of extension domains for classical Besov spaces \(B^s_{p,q}\), \(0<s<1\), \(0<p<\infty \), \(0<q\le \infty \), defined via the \(L^p\)-modulus of smoothness of a function.     

Injectivity Conditions for Some Classes of Domains. I

New conditions are given for continuously differentiable mappings of some plane domains to be injective. In the case of a circular plane domain, these criteria coincide with well-known conditions. The method of locally homeomorphic extension is used. Bibliography: 18 titles.     

Sobolev functions on infinite-dimensional domains

We study extensions of Sobolev and BV functions on infinite-dimensional domains. Along with some positive results we present a negative solution of the long-standing problem of existence of Sobolev extensions of functions in Gaussian Sobolev spaces from a convex domain to the whole space.     

Second-order properly elliptic boundary value problems on irregular plane domains

Conditions are presented under which properly elliptic second-order boundary value problems are well posed on irregular plane domains. The coefficients can be discontinuous. The results include known results for coercive forms, and also reduce to known results on proper ellipticity when the coefficients and domain are smooth. The main tool is an “inverse five-lemma” which relates the Neumann problem on a plane domain to a related modified Dirichlet problem. This inverse five-lemma can be used in a variety of settings. We show how it can be used to translate results of Grisvard on the index of Dirichlet operators in Sobolev spaces $\text{H}{}^{\mathrm{s}}\text{(Ω)}$ to results on Neumann operators, and examine the implications for regularity.     

Admissible Domains for Higher Order Differential Equations

This paper analyzes the geometric structure of certain domains in the complex plane which arise in the asymptotic theory of linear ordinary differential equations containing a parameter. These domains, called admissible, are domains in which an asymptotic representation of the solution of the differential equation may be found and across whose boundaries these representations may undergo a rapid change of asymptotic behavior (the Stokes phenomenon). A knowledge of the disposition of those domains associated with a particular differential equation is necessary for a satisfactory asymptotic theory of the equation. The main analysis gives necessary and sufficient conditions for identifying admissible domains and gives a procedure for obtaining particular admissible subdomains of a given domain. Sufficient conditions are established to determine the maximality of admissible domains. A section of examples is included to highlight the salient features of this theory. In all of the results, criteria involving only purely local properties of the boundary are needed to determine the global properties of admissibility and maximal admissibility .     

Szegö and Bergman projections on non-smooth planar domains

We establish L^p regularity for the Szegö and Bergman projections associated to a simply connected planar domain in any of the following classes: vanishing chord arc; Lipschitz; Ahlfors-regular; or local graph (for the Szegö projection to be well defined, the local graph curve must be rectifiable). As applications, we obtain L^p regularity for the Riesz transforms, as well as Sobolev space regularity for the non-homogeneous Dirichlet problem associated to any of the domains above and, more generally, to an arbitrary proper simply connected domain in the plane.     

Generalized Grunsky coefficients and inequalities

The generalized Grunsky coefficients are defined in this paper for all locally univalent meromorphic functions in any domain in the complete complex plane. Various explicit formulas for these coefficients are established. Necessary conditions for univalence are obtained in arbitrary domains and in the unit disc in particular. The first one generalizes Grunsky inequalities and the second one is an extension of the Nehari-Schwarzian derivative condition.     

The extension of functions of Sobolev classes beyond the boundary of the domain on Carnot groups

We prove the theorem on extension of the functions of the Sobolev space W _p ^l (Ω) which are defined on a bounded (ε, δ)-domain Ω in a two-step Carnot group beyond the boundary of the domain of definition. This theorem generalizes the well-known extension theorem by P. Jones for domains of the Euclidean space.     

Complexity in Complex Analysis

We show that the classical kernel and domain functions associated to an n-connected domain in the plane are all given by rational combinations of three or fewer holomorphic functions of one complex variable. We characterize those domains for which the classical functions are given by rational combinations of only two or fewer functions of one complex variable. Such domains turn out to have the property that their classical domain functions all extend to be meromorphic functions on a compact Riemann surface, and this condition will be shown to be equivalent to the condition that an Ahlfors map and its derivative are algebraically dependent. We also show how many of these results can be generalized to finite Riemann surfaces.     

The Cauchy problem in Sobolev spaces for Dirac operators

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary-value problems. We obtain the necessary and (separately) sufficient conditions for the solvability of the Cauchy problem for a Dirac operator A in Sobolev spaces in a bounded domain D ? ? ⁿ with a piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of harmonic extension from a smaller domain to a larger one. Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function u in a Sobolev space H ^s (D), s ∈ ?, from its values on Γ and values Au in D, where Γ is an open connected subset of the boundary ?D. It is worth pointing out that we impose no assumptions about geometric properties of the domain D, except for its connectedness.     

Bargaining with nonanonymous disagreement: Decomposable rules

We analyze bargaining situations where the agents’ payoffs from disagreement depend on who among them breaks down the negotiations. We model such problems as a superset of the standard domain of Nash (1950). We first show that this domain extension creates a very large number of new rules. In particular, decomposable rules (which are extensions of rules from the Nash domain) constitute a nowhere dense subset of all possible rules. For them, we analyze the process through which “good” properties of rules on the Nash domain extend to ours. We then enquire whether the counterparts of some well-known results on the Nash (1950) domain continue to hold for decomposable rules on our extended domain. We first show that an extension of the Kalai–Smorodinsky bargaining rule uniquely satisfies the Kalai and Smorodinsky (1975) properties. This uniqueness result, however, turns out to be an exception. We characterize the uncountably large classes of decomposable rules that survive the Nash (1950), Kalai (1977), and Thomson (1981) properties.